Quantum simulation of exotic $\mathcal{PT}$-invariant topological nodal loop bands with ultracold atoms in an optical lattice

Since the well-known $\mathcal{PT}$ symmetry has its fundamental significance and implication in physics, where $\mathcal{PT}$ denotes a joint operation of space inversion $\mathcal{P}$ and time reversal $\mathcal{T}$, it is important and intriguing to explore exotic $\mathcal{PT}$-invariant topological metals and to physically realize them. Here we develop a theory for a different type of topological metals that are described by a two-band model of $\mathcal{PT}$-invariant topological nodal loop states in a three-dimensional Brillouin zone, with the topological stability being revealed through the $\mathcal{PT}$-symmetry-protected nontrivial ${\mathbb{Z}}_2$ topological charge even in the absence of both $\mathcal{P}$ and $\mathcal{T}$ symmetries. Moreover, the gapless boundary modes are demonstrated to originate from the nontrivial topological charge of the bulk nodal loop. Based on these exact results, we propose an experimental scheme to realize and to detect tunable $\mathcal{PT}$-invariant topological nodal loop states with ultracold atoms in an optical lattice, in which atoms with two hyperfine spin states are loaded in a spin-dependent three-dimensional optical lattice and two pairs of Raman lasers are used to create out-of-plane spin-flip hopping with site-dependent phase. It is shown that such a realistic cold-atom setup can yield topological nodal loop states, having a tunable band-touching ring with the twofold degeneracy in the bulk spectrum and nontrivial surface states. The nodal loop states are actually protected by the combined $\mathcal{PT}$ symmetry and are characterized by a ${\mathbb{Z}}_2$-type invariant (or topological charge), i.e., a quantized Berry phase. Remarkably, we demonstrate with numerical simulations that (i) the characteristic nodal ring can be detected by measuring the atomic transfer fractions in a Bloch-Zener oscillation; (ii) the topological invariant may be measured based on the time-of-flight imaging; and (iii) the surface states may be probed through Bragg spectroscopy. The present proposal for realizing topological nodal loop states in cold-atom systems may provide a unique experimental platform for exploring exotic $\mathcal{PT}$-invariant topological physics.

Figure 1

An NL in three-dimensional momentum space, enclosed by a tiny circle.

Figure 2

An NL in a three-dimensional Brillouin zone, enclosed by different loops.

Figure 3

The proposed cold-atom system and laser configuration. (a) The cubic optical lattice and the effective atomic hopping configuration between nearest-neighbor lattice sites. It contains the spin-conserved hopping $t_{\uparrow }$ along each axis, $t_{\downarrow }$ in the $xy$ plane but $-t_{\downarrow }$ out of the plane, the spin-flip hopping from $\left|\downarrow \rangle \right.$ to $\left|\uparrow \rangle \right.$, which is $\pm t_{\mathrm{so}}$ staggered along the $z$ axis, and its Hermitian conjugation progress. (b) The spin-dependent lattice structure and laser configuration of the Raman transition between spin states $\left|\uparrow \rangle \right.$ and $\left|\downarrow \rangle \right.$ along the $z$ axis. The effective Raman field strength proportional to $sin\left(k_{0}z\right)$ (green dotted line) consists of two pairs of plane waves denoted by their Rabi frequencies ${\mathrm{\Omega }}_{1,\pm }$ (which gives the laser field ${\mathrm{\Omega }}_1$) and ${\mathrm{\Omega }}_{2,\pm }$ (which gives the laser field ${\mathrm{\Omega }}_2$). Two spin states are coupled via a two-photon Raman transition with a large detuning ${\mathrm{\Delta }}_d$ from an excited state $|e\rangle$ and a two-photon detuning $\delta$.

Figure 4

Phase diagram and tunable NL states. (a) Phase diagram. III, II.a, II.b, and I correspond to a trivial insulator, an NL on the $k_z=0$ plane, an NL on the $k_z=\pi /a$ plane, and two coexisting NLs on the two planes, respectively. (b) The evolution of an NL during the increase of $\beta$ is denoted by the black arrows in (a). (c) The distribution of the topological numbers of one-dimensional subsystems on the $k_x-k_y$ plane in difference phases during the increase of $\beta$ is marked by the white arrow in (a).

Figure 5

The bulk gap with the gap-closing points forming a nodal ring and its detection. (a) A nodal ring on the $k_z=0$ plane with the parameter $m_z=2.0$. (b) A nodal ring on the $k_z=\pi /a$ plane with the parameter $m_z=-1.8$. (c,d) The momentum distribution of the atomic transfer fraction ${\xi }_z(k_x,k_y)$ with the parameter $F=0.2$. The maximum positions form a curve that reveals the nodal ring in (a) and (b), respectively. Other parameters in (a)–(d) are $t_{\uparrow }=1$ as the energy unit, $t_{\downarrow }=t_{\mathrm{so}}=0.3$.

Figure 6

(a) The Berry phase $\gamma (k_x,k_y)$ calculated by using the Bloch Hamiltonian (21). (b) The simulated measurement of Berry phase $\gamma (k_x,k_y)$ with the lattice size $32\times 32\times 32$ and an additional harmonic trap with the parameter $\nu =0.001$ (see the text) by using Eq. (28). The black solid line denotes the nodal loop in this case. (c) The energy spectrum with respect to $k_x$ for fixed $k_y=0$ and open boundary condition along the $z$ axis, and the surface states inside the gap. (d) The density distribution of two surface modes at $k_x=k_y=0$. Other parameters in (a)–(d) are $t_{\uparrow }=1$ as the energy unit, $t_{\downarrow }=t_{\mathrm{so}}=0.3$, and $m_z=2.0$.